Theorem sbcel12g 2921

Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
sbcel12g	|- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )

Proof of Theorem sbcel12g

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2818	. . 3 |- ( z = A -> ( [ z / x ] B e. C <-> [. A / x ]. B e. C ) )
2		dfsbcq2 2818	. . . . 5 |- ( z = A -> ( [ z / x ] y e. B <-> [. A / x ]. y e. B ) )
3	2	abbidv 2196	. . . 4 |- ( z = A -> { y | [ z / x ] y e. B } = { y | [. A / x ]. y e. B } )
4		dfsbcq2 2818	. . . . 5 |- ( z = A -> ( [ z / x ] y e. C <-> [. A / x ]. y e. C ) )
5	4	abbidv 2196	. . . 4 |- ( z = A -> { y | [ z / x ] y e. C } = { y | [. A / x ]. y e. C } )
6	3, 5	eleq12d 2149	. . 3 |- ( z = A -> ( { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } <-> { y | [. A / x ]. y e. B } e. { y | [. A / x ]. y e. C } ) )
7		nfs1v 1856	. . . . . 6 |- F/ x [ z / x ] y e. B
8	7	nfab 2223	. . . . 5 |- F/_ x { y | [ z / x ] y e. B }
9		nfs1v 1856	. . . . . 6 |- F/ x [ z / x ] y e. C
10	9	nfab 2223	. . . . 5 |- F/_ x { y | [ z / x ] y e. C }
11	8, 10	nfel 2227	. . . 4 |- F/ x { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C }
12		sbab 2205	. . . . 5 |- ( x = z -> B = { y | [ z / x ] y e. B } )
13		sbab 2205	. . . . 5 |- ( x = z -> C = { y | [ z / x ] y e. C } )
14	12, 13	eleq12d 2149	. . . 4 |- ( x = z -> ( B e. C <-> { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } ) )
15	11, 14	sbie 1714	. . 3 |- ( [ z / x ] B e. C <-> { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } )
16	1, 6, 15	vtoclbg 2659	. 2 |- ( A e. V -> ( [. A / x ]. B e. C <-> { y | [. A / x ]. y e. B } e. { y | [. A / x ]. y e. C } ) )
17		df-csb 2909	. . 3 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
18		df-csb 2909	. . 3 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
19	17, 18	eleq12i 2146	. 2 |- ( [_ A / x ]_ B e. [_ A / x ]_ C <-> { y | [. A / x ]. y e. B } e. { y | [. A / x ]. y e. C } )
20	16, 19	syl6bbr 196	1 |- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   [wsb 1685   {cab 2067   [.wsbc 2815   [_csb 2908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816  df-csb 2909

This theorem is referenced by:  sbcnel12g  2923  sbcel1g  2925  sbcel2g  2927  sbccsb2g  2935